An observatory is to be in the form of a right circular cylinder surmounted by a hemispherical dome. If the hemispherical dome costs 6 times as much per square foot as the cylindrical wall, what are the most economic dimensions for a volume of 16000 cubic feet?
The radius of the cylindrical base (and of the hemisphere) is _____ ft. (Round to the nearest tenth).
Here are some questions to help guide you. What are the equations of surface area for the lateral surface area of a cylinder, and that of a hemisphere? What would be the cost of the cylindrical part if the cost was C per ft^2 (if you need more help: you want to buy 2 kg of apples and the price is $3 per kg: what do you do to these two numbers to figure out the total cost?) Post your answers and further questions below and we'll go from there.
π = piOriginally Posted by majamin
Area: A = 2πr2+2πrh
C = (6)2πr2+2πrh
Cost: C = (32000/r) + (72πr2/3)
This could be wrong.. I'm not very good with these word problems. Are these equations correct?
Good start. Because we are just calculating the wall of a cylinder the equation has to change to(it doesn't include
). The area of a hemisphere is half of
or
The cost will be
. I getCost: C = (32000/r) + (72πr2/3)
This could be wrong.. I'm not very good with these word problems. Are these equations correct?where C is the cost as above. We need to minimize cost, so this is an opportune time for the derivative (I'll let you try to fill in the details). As a final answer, I get
.
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